We provide an Information-Geometric formulation of accelerated natural gradient on the Riemannian manifold of probability distributions, which is an affine manifold endowed with a dually-flat connection. In a non-parametric formalism, we consider the full set of positive probability functions on a finite sample space, and we provide a specific expression for the tangent and cotangent spaces over the statistical manifold, in terms of a Hilbert bundle structure that we call the Statistical Bundle. In this setting, we compute velocities and accelerations of a one-dimensional statistical model using the canonical dual pair of parallel transports and define a coherent formalism for Lagrangian and Hamiltonian mechanics on the bundle. We show how our formalism provides a consistent framework for accelerated natural gradient dynamics on the probability simplex, paving the way for direct applications in optimization.

我们在概率分布的黎曼流形上提供了加速自然梯度的信息几何公式，黎曼流形是一个具有对偶平面连接的仿射流形。在非参数形式主义中，我们考虑有限样本空间上的全套正概率函数，并根据我们称为统计包。在此设置中，我们使用典型的双平行传输对计算一维统计模型的速度和加速度，并为束上的拉格朗日和哈密顿力学定义相干形式。